Let's say we have an equation $y = \beta_0 + \beta_1 x_1 + \dots + \beta_p x_p$, where the unknowns are $(x_1, \dots, x_p,y)$. It is said that this describes a hyperplane $H$ of the affine space $\mathbb{R}^{p + 1}$. How is this a hyperplane of the affine space $\mathbb{R}^{p + 1}$? What is it about the mathematics of affine spaces that makes this so?
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Affine:
Affine is a term to say it not a linear function, but shifted/translated( do not confuse with a nonlinear function either). Example $y = ax$ is linear as it includes the $0$ point. It(Linear) accepts the principle of superposition and homogeneity, whereas $y = ax + b$ is not a linear function but an affine function.
For your function the constant term $\beta_{0}$ makes it an affine function.
Next Hyperplane,
Hyperplane or a Hypersurface is a subspace of $n$ - dimensional space with one dimension less. Normally its something that splits an $n$ dimensional space.
General examples that are shown are.
"a point" is a hyperplane in 1-D space $\mathbb{R}^{1}$. It splits the line.
"a line" is a hyperplane in 2-D space $\mathbb{R}^{2}$. It splits the plane.
"a plane" is a hyperplane in 3-D space $\mathbb{R}^{3}$.
also an unit sphere is a "hypersurface" or a 2-D object in a 3-D space
Other way to see may be is , how many variables you require to define a point in a surface(in general)/plane. Always it is one dimension less.
So if a plane is defined as $Ax +By + Cy +d$ then $x,y,z$ is the space $\mathbb{R}^{3}$ plus a shift/translation/a constant term that is "real value" makes it to $\mathbb{R}^{3+1}$ which is an affine space(not $\mathbb{R}^{4}$, . So what above describes is a "hyperplane" a plane of one dimension less in that affine space.
So in general an affine space is defined by $\mathbb{R}^{n+1}$
If you don't have the constant term, then it is a plane containing the origin and you can relax mentioning it as an affine space.
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